While waiting for our carry-out food to be ready at El Norteño, the neighborhood Salvadoran restaurant, R and I were trying to crack a math problem he had struggled with for over an hour: to completely factor x4 + 64. R was clearly getting frustrated, as is his wont: he's persistent and somewhat emotional when he cannot do a job the way he wants to do it (or feels that it should be done). I used some tricks that his book hadn't yet introduced to get the answer, (x+2+2i)(x-2+2i)(x+2-2i)(x-2-2i), but of course that didn't satisfy him. The book hinted that one should add and subtract 16x2 from the original polynomial, but neither R nor I could see how that might help.
So we stared at the problem and ate chips with salsa. I tried working backwards from the answer, getting the intermediate result (x2+4x+8)(x2-4x+8), which we still couldn't see how to connect up to the starting point. R was getting downright mad. He studied our scribblings for another five minutes, and then laughed and said, "It's obvious!" He had figured out the right grouping of terms, quite a counter-intuitive one.
I had to laugh too, because there are so many things like that: "obvious" only after hard work or bitter experience. I imagined struggling with a problem not for an hour, but for months or years — and not with a known answer, nor with any hints, but with only the vague hope that a solution exists ... like Copernicus trying to derive the orbit of Mars (it took him a decade) or Wiles's proof of Fermat's Last Theorem.
What kind of faith must one have to win through near-impenetrable thickets of complexity to achieve the (after-the-fact) "obvious"?
Saturday, January 29, 2000 at 08:54:57 (EST) = 2000-01-29
TopicScience - TopicPersonalHistory
There are some famous stories about relative 'obviousness' in mathematics. In one book of theorems published by a now-reknown mathematician, the phrase 'It is obvious that ...', or something like it, turned up frequently; and one person (another known mathematician, but a recent one) was quoted to say something resembing, "Whenever I saw that phrase, I knew I had several hours of hard math ahead of me!"
[[RadRob]] (the 'R' from above), October 4, 2001(correlates: SubtopicTolkien, NoProblem, GoodMistakes, ...)